Internal problem ID [8501]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 921.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_symmetry_[F(x),G(x)*y+H(x)]]]
Solve \begin {gather*} \boxed {y^{\prime }+\left (-\frac {\ln \relax (y)}{x}+\frac {\ln \relax (y)}{x \ln \relax (x )}-\textit {\_F1} \relax (x )\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 30
dsolve(diff(y(x),x) = -(-1/x*ln(y(x))+1/x/ln(x)*ln(y(x))-_F1(x))*y(x),y(x), singsol=all)
\[ y \relax (x ) = {\mathrm e}^{\frac {x c_{1}}{\ln \relax (x )}} {\mathrm e}^{\frac {x \left (\int \frac {\textit {\_F1} \relax (x ) \ln \relax (x )}{x}d x \right )}{\ln \relax (x )}} \]
✓ Solution by Mathematica
Time used: 0.228 (sec). Leaf size: 91
DSolve[y'[x] == (F1[x] + Log[y[x]]/x - Log[y[x]]/(x*Log[x]))*y[x],y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^x\left (\frac {\log (y(x))-\log (K[1]) \log (y(x))}{K[1]^2}-\frac {\text {F1}(K[1]) \log (K[1])}{K[1]}\right )dK[1]+\int _1^{y(x)}\left (\frac {\log (x)}{x K[2]}-\int _1^x\frac {\frac {1}{K[2]}-\frac {\log (K[1])}{K[2]}}{K[1]^2}dK[1]\right )dK[2]=c_1,y(x)\right ] \]